Representation Theory A

An introduction to quiver representations, derived categories, and Hall algebras for master's-level students.

Instructor: Francesco Sala

Term: 2025/26

Course Overview

Following Kac and Moody, a diagram can be used to define a Lie algebra. For instance, the \(A_N\) Dynkin diagram corresponds to the special linear Lie algebra \(\mathfrak{sl}(N+1)\). By orienting the diagram, we obtain a quiver, which in turn defines an abelian category of quiver representations. This raises a natural question: how do the “algebraic” properties of the Lie algebra relate to the “categorical” properties of its corresponding representations?

A first answer is given by Gabriel’s theorem. It establishes a fundamental link by creating a bijection between the positive roots of the Lie algebra and the isomorphism classes of indecomposable representations of the quiver. These classes form a basis for the Grothendieck group of the category.

However, the Grothendieck group is only a “shadow” of the category, discarding rich information about morphisms and extensions. To establish a deeper connection that involves the category’s full structure, one must turn to Hall algebras. The Hall algebra is constructed directly from the extension structure within the abelian category. This powerful framework reveals a profound relationship: the Hall algebra of a quiver is directly related to a (quantum) deformation of the universal enveloping algebra of the corresponding Lie algebra. This provides a “categorification” of the Lie algebra, where the algebra’s structure is realized through the “categorical” structure of its representations.

This course offers an introduction to the representation theory of quivers and the theory of Hall algebras. The course is designed for master’s-level students with a background in linear algebra and basic homological algebra.

We will begin with a study of quiver representations, including quiver representations and path algebras. We will explore Gabriel’s theorem, which classifies quivers of finite representation type, and delve into the geometric aspects of quiver representations.

The second part of the course introduces derived categories and derived functors, culminating in a presentation of Beilinson’s theorem. This result establishes an explicit relation between the bounded derived category of projective space and the bounded derived category of a quiver with relations.

The third part offers a concise introduction to Hall algebras. We will define Hall algebras associated with abelian categories, with particular emphasis on those arising from quivers. This perspective connects the two central themes of the course, showing how the combinatorial structure of quiver representations gives rise to rich algebraic structures. We will also discuss Cramer’s theorem, which relates Hall algebras to derived equivalences.

Prerequisites

Linear algebra and basic homological algebra (Ext groups). Familiarity with basic category theory is helpful but not required.

Primary Texts

Quiver representations

  • A. Assem, D. Simson, and A. Skowroński, Elements of the Representation Theory of Associative Algebras, Volume 1: Techniques of Representation Theory, Cambridge University Press, 2006. A systematic introduction from the perspective of associative algebras and their module categories.
  • W. Crawley-Boevey, Lectures on Representations of Quivers, 1992, and R. Schiffler, Quiver Representations, Springer, 2014. More accessible introductions.

Derived categories

  • D. Miličić, Lectures on Derived Categories, 2014, and S. I. Gelfand and Y. I. Manin, Methods of Homological Algebra, Springer, 2003. Comprehensive overviews.
  • D. Huybrechts, Fourier–Mukai Transforms in Algebraic Geometry, Oxford Science Publications, 2006. A gentler introduction.

Hall algebras

Possible Topics for the Final Exam

Braid groups and derived equivalences

  • Y. Sekiya and K. Yamaura, Tilting theoretical approach to moduli spaces over preprojective algebras. Published version
  • P. Seidel and R. Thomas, Braid group actions on derived categories of coherent sheaves. Published version
  • A. Ishii and H. Uehara, Autoequivalences of derived categories on the minimal resolutions of (A_n)-singularities on surfaces. Published version

Bridgeland’s stability conditions

  • T. Bridgeland, Stability Conditions on Triangulated Categories. Published version
  • T. Bridgeland, Stability Conditions and Kleinian Singularities. arXiv version

Derived McKay equivalence

Derived categories in the topological setting

  • M. Kashiwara and P. Schapira, Sheaves on Manifolds.

Joyce’s Hall algebra

  • D. Joyce, Configurations in abelian categories—I: Basic properties and moduli stacks. Published version
  • D. Joyce, Configurations in abelian categories. II: Ringel–Hall algebras. Published version
  • T. Bridgeland, An introduction to motivic Hall algebras. Published version

Hall algebras of quivers

  • Ringel and Green’s theorems on Hall algebras of quivers without edge-loops, following Olivier Schiffmann’s lecture notes.
  • S.-J. Kang, Ringel–Hall algebra construction of quantum Borcherds–Bozec algebras. Published version

Hall algebras of curves

  • I. Burban and O. Schiffmann, On the Hall algebra of an elliptic curve, I. Published version
  • O. Schiffmann and E. Vasserot, Hall algebras of curves, commuting varieties and Langlands duality. Published version

Hall algebras and derived categories